Translation of Mathematica scripts of xah lee

and some extra figures

http://xahlee.info/MathGraphicsGallery_dir/Transform2DPlot_dir/transform2DPlot.html
The transform_2d_plot() was generated by claude 3.7 after prompted to recreate xah's mathematica function

The function plots a 2D transformation: $$ (x, y) \to (x', y') \text{ where } (x', y') = transform\_func(x, y)$$
The x and y grid values are defined over $$[x_{\text{min}}, x_{\text{max}}] $$ and $$ [y_{\text{min}}, y_{\text{max}}], \text{ respectively.} $$
For each grid point (x, y), the transformation is applied, and the transformed coordinates (x', y') are plotted.
The color is determined by a default function or a user-defined function, such as: hsv_to_rgb(theta, 0.8, 0.9)

$$ \text{Rotation Transform: Applies a rotation matrix to } (x, y) \text{ based on distance from origin.} $$ $$ \text{If } r < 1, \text{ rotates by } \theta = (1 - r^2) \cdot n \text{ and scales by } r^2. $$

This transform rotates points around the origin, with the rotation angle depending on the radial distance $r$. For points closer to the origin $(r < 1)$, the rotation is stronger, and the points are also scaled by $r^2$. This creates a swirling effect that intensifies near the center.

$$ \text{Sine Ripple Transform: Adds a sinusoidal ripple effect to } (x, y) \text{ based on radial distance.} $$ $$ \text{Transforms } (x, y) \text{ as } (x + \sin(r) \cdot \frac{x}{r}, y + \sin(r) \cdot \frac{y}{r}). $$

This transform adds a sinusoidal ripple effect to the coordinates, where the amplitude of the ripple depends on the radial distance $r$. The effect is stronger for larger $r$, creating wave-like distortions that radiate outward from the origin.

$$ \text{Parametric Tangent Transform: Applies a hyperbolic tangent-like transformation.} $$ $$ \text{Transforms } x \text{ as } \frac{3^2 \cdot x}{2^2 \cdot y} \text{ and leaves } y \text{ unchanged.} $$

This transform applies a hyperbolic tangent-like transformation to the $x$-coordinate while leaving the $y$-coordinate unchanged. The result is a stretching or compressing effect along the $x$-axis, depending on the value of $y$.

$$ \text{Sine Transform: Maps } (x, y) \text{ using trigonometric functions.} $$ $$ \text{Transforms } (x, y) \text{ as } (\cos(y), \sin(x)). $$

This transform maps the coordinates using trigonometric functions, creating a wave-like pattern. The $x$-coordinate is replaced with $cos(y)$, and the $y$-coordinate is replaced with $\sin(x)$. This results in a complex, oscillating distortion.

$$ \text{Polynomial Transforms: Applies polynomial mappings to } (x, y). $$ $$ \text{For example, } (y \cdot x, x^2) \text{ or } (y \cdot x, x^3). $$

These transforms apply polynomial functions to the coordinates, such as multiplying (x) and (y) or raising (x) to a power. The resulting distortions can create curves, folds, or other nonlinear effects depending on the polynomial used.

$$ \text{Trigonometric Transforms: Combines trigonometric functions with } (x, y). $$ $$ \text{For example, } (\sin(y) \cdot \cos(y), x \cdot y) \text{ or } (\tan(y), x \cdot y). $$

These transforms combine trigonometric functions (e.g., sine, cosine, tangent) with the coordinates. The result is a complex, periodic distortion that can create intricate patterns or waves.

$$ \text{Exponential Transform: Uses exponential functions to transform } (x, y). $$ $$ \text{Transforms } (x, y) \text{ as } (x \cdot y, \exp(x)). $$

This transform applies exponential functions to the coordinates, such as $\exp(x)$. The result is a rapid scaling or distortion that intensifies for larger values of $x$, creating dramatic effects.

$$ \text{Zeta Transform: Applies the Riemann zeta function to } (x, y). $$ $$ \text{Transforms } (x, y) \text{ as } (x \cdot y, \text{Re}(\zeta(x))). $$

This transform uses the Riemann zeta function $\zeta(x)$ to modify the coordinates. The real part of $\zeta(x)$ is applied to the $y$-coordinate, creating a distortion that reflects the complex behavior of the zeta function.

$$ \text{Mandelbrot Transform: Iterates complex numbers to generate fractal patterns.} $$ $$ \text{Transforms } (x, y) \text{ based on escape time and angle of complex iteration.} $$

This transform uses the Mandelbrot set algorithm to generate fractal patterns. The coordinates are iterated as complex numbers $z = z^2 + c$, and the transformation depends on the escape time and angle of iteration.

$$ \text{Julia Transform: Similar to Mandelbrot but with a fixed complex constant } c. $$ $$ \text{Transforms } (x, y) \text{ based on escape time and angle for } z = z^2 + c. $$

This transform is similar to the Mandelbrot transform but uses a fixed complex constant $c$. The resulting patterns are Julia sets, which are fractals with intricate, self-similar structures.

$$ \text{Normalization Transforms: Normalizes } (x, y) \text{ by radial distance.} $$ $$ \text{For example, } \left(\frac{x}{r}, \frac{y}{r}\right) \text{ or } \left(\frac{x \cdot \log(r)}{r}, \frac{y \cdot \log(r)}{r}\right). $$

These transforms normalize the coordinates by dividing by the radial distance $r$. This can create a radial scaling effect, with optional logarithmic adjustments for more complex distortions.